RC Circuit Transients

Experiment: Transient Response of an RC Circuit

1. Aim

To study the charging and discharging behavior of a capacitor in a series RC circuit and to determine the Time Constant ($\tau$) by fitting the experimental data.

2. Apparatus / Components Required

• SEELab3 or ExpEYES-17 unit
• One Resistor ($R = 1\text{ k}\Omega$ )
• One Capacitor ($C = 1\text{ }\mu F$ )
• Connecting wires
• PC or Smartphone with SEELab3 software

3. Theory & Principle

When a voltage step ($V_0$) is applied to an empty capacitor through a resistor, the voltage across the capacitor ($V_C$) does not jump instantly. It increases exponentially according to: \(V_C(t) = V_0(1 - e^{-t/RC})\)

Similarly, when a charged capacitor is allowed to discharge through a resistor, the voltage decays as: \(V_C(t) = V_0 e^{-t/RC}\)

The product $RC$ is called the Time Constant ($\tau$). It represents the time required for the capacitor to charge to approximately $63.2\%$ of the applied voltage or discharge to $36.8\%$ of its initial value.

4. Circuit Diagram / Setup

1. Series Connection: Connect the Resistor ($R$) and Capacitor ($C$) in series.
2. Input Source: Connect the free end of the resistor to OD1 (Digital Output used for the voltage step).
3. Ground: Connect the free end of the capacitor to GND.
4. Measurement: Connect the junction between the resistor and capacitor to A1 to monitor $V_C$.

5. Procedure

1. Open the SEELab3 app and select the “RC Transient” experiment.
2. The software is programmed to toggle OD1 from $0V$ to $5V$ and simultaneously start high-speed data acquisition on A1.
3. Click on “Charge” (or Start). The software will apply the step and display the rising exponential curve.
4. Click on “Discharge”. The software will set OD1 to $0V$ and capture the falling exponential curve.
5. Data Fitting: Select a region of the captured curve. Use the “Fit” tool to apply an exponential fit. The software will display the value of the time constant ($RC$).
6. Compare the experimental $RC$ value with the theoretical value calculated from the component markings ($R \times C$).

6. Observation Table

7. Results and Discussion

• The voltage across the capacitor followed an exponential path during both charging and discharging phases.
• The measured time constant was found to be ____ ms.
• The results verify that it takes approximately $5\tau$ for the capacitor to reach its steady-state (fully charged or discharged) condition.

8. Precautions

1. Selection of RC: Choose $R$ and $C$ values such that the time constant is between $1\text{ ms}$ and $100\text{ ms}$. If $\tau$ is too small, the software may not have enough sampling resolution to capture the curve accurately.
2. Input Impedance: The $1\text{ M}\Omega$ input impedance of A1 is in parallel with the capacitor during discharge. For very high $R$ values (e.g., $1\text{ M}\Omega$), this will introduce significant error.
3. Residual Charge: Ensure the capacitor is fully discharged before starting a new “Charge” cycle for a clean plot starting from $0V$.

9. Troubleshooting